Inserted: 7 jan 2014
Last Updated: 9 mar 2016
Journal: Arch. Ration. Mech. Anal.
We introduce a new method to prove the isoperimetric property of the ball for the first eigenvalue of the Robin-Laplacian. Our technique applies to a full range of Faber-Krahn inequalities in a nonlinear setting and for non smooth domains, including the open case of the torsional rigidity. The analysis is based on regularity issues for free discontinuity problems in spaces of functions of bounded variation. As a byproduct, we obtain the best constants for a class of Poincaré inequalities with trace terms in $R^N$.